June 6, 1949.
Dear Kim:

Since our very provocative discussion on the mltiple-target theory,
I have had a chance to look up some of the literature, ani am surprised
to have to confess that there has been surprisingly little treatment.
However, I still think that something might be found e.g., as a tresament
eof the sigmoid survival curves when bacteria are allowed to form aicro-
Colonies on agar before being irradiated. Luria and Dulbecco's approach
is substantially the same, although they have to use certain slightly
different terms, and appear to have dapended on an arithmetic summation of
their series in evaluating the numbers of units. Luria and Latarjet's
J. Bact. paper on irradiation of infected bacteria was the only clear statement
that I could find along the lines of the theoryathat you are developing,
and they seem to have used empirical methods of fitting to the untrane-
formed curves. While on the train, I tried to see what I could do, but
didn't come out with very auch. Assuming a constant nusber of nuclei,a,
we have, of course:

f2) P= L- (e4)®, op og= (1- eM) 8,

I don't see any way of almplifying this to facilitate the estimation of
a and no from the p/d data, except possibly to approximate, for ad large with
respect to a:

§2) & = ne, This just means, what we know already, that the

log p/ d curves will become asymptotic, for large doses to lines with sl
-~a Teen will extrapolate to the p - 0 line with an intercept dose equivalent
to logn/a.

imria and Latarjet refer to Delbrick's derivation of the expression:

(3) pi eA nad which refere to the "apparent survival" using the full
Poisson distribution (untruncséed). I assume that this is the function which
transformls so nicely with loglogs:

(4) legleg 1/p = logn~ ad, ‘There is one major difficulty with this
expression that I can't see the solution for, ani wonder how you ney ave handled
p

it. Because of the derivation Broa a fictitious Poisson, the does not
refer to p/p. where these er to the o ues with and without

radiation,“buf to p Po = P,/P, + e~" , It te fairly obvious in (4) that

p does not Usepme unity when no d is delivered. The full expression should read,

then, =f
5) loglog (140 )/g, = log n - ad, which would not give precise
tr&ight lines when loglog p, isfplotted against 4.
The correction for p will become negligible when e~? is small (e.g. will be

less than 1% for n more than 5), and for values of @ which allow low survival
may be unimportant even for small n. But in the first couple of decaies of __

_ ‘killing, with values of n ca. 2 or 3, I think that this theory demands a

rather appreciable deviation from linearity. However, the expression should

lend itself to solution by successive approximation, by estimating an uncorrected
n from (4), and then substit&ting this value of n in (5) and so on,.

I haven't been able to find that reference to Yale's paper, as we don't have
a file of the Proc. Roy Stat. Soe London here, but I have a rather distinct recol-
lection thit Lt was about 1916, and that it covered a good approximate function
4nstead of (1), using tables of the gamma-funetions. It doesn'$ really have
mach bearing on the problem of (4). I'LL be very mach interested to hear how
your analysis comprres with this mne, and am Looking forward to seeing your manuscript.
If you can't find the Yule refereamce. Dr. Rob't Boche, Inetitute of Radiobiology
& Biophysics, U. Chiearc, Shi. 27, is when I heard shout ft from in the first

piace.

Itm not sending the "analysis" of the N. tetrasperma data, as I found an
error in it: I neglente? to Inelude tha 32-bit cless. You'l]. be interested to
Look up Uber and @oddard, JGenPhysiol 17. 597, and recalculate their data on
the basis that the 4etarget—killad are dead, and all of the 3+, and half of the
2 class are salf-sterile.

That stuff of yours on induced belanced Neterokaryons in Neurospora sounds

very exciting, and Tam sofng to help myself to the ideas it provoked concerning
some parallel expts. in diploid KeL2,

Sincerely,

Joshua Laderberg
P.S. I didn't mind ny p's and q's too carefully on the first page. But I think
that you can get what I mean without revising it any further than I did in ink..

Notlee that (2) gives you log p - log n - ad, while (4) gives you

loglog (1/l-p) = log n~ ad. That is, innthe ting case,

p should approximate Log (1/lep)., Liee, e? = 1 ~ Pp (4 ff =f Blewee)
which is of course true for small values of p.
4 = ()-e-*)

~~ fe

(1- )" = = ,9%03
l-ne~ = r- lo-.ol = ,900 oX
pwr
wll hare sn

 

fu —->